Integrand size = 35, antiderivative size = 50 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=a x+\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^{1+n}}{1+n} \] Output:
a*x+1/2*b*x^2+1/3*c*x^3+(a*x+1/2*b*x^2+1/3*c*x^3)^(1+n)/(1+n)
Time = 0.51 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\frac {x (6 a+x (3 b+2 c x)) \left (1+n+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right )}{6 (1+n)} \] Input:
Integrate[(a + b*x + c*x^2)*(1 + (a*x + (b*x^2)/2 + (c*x^3)/3)^n),x]
Output:
(x*(6*a + x*(3*b + 2*c*x))*(1 + n + (a*x + (b*x^2)/2 + (c*x^3)/3)^n))/(6*( 1 + n))
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2024, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a+b x+c x^2\right ) \left (\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n+1\right ) \, dx\) |
\(\Big \downarrow \) 2024 |
\(\displaystyle \int \left (\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n+1\right )d\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^{n+1}}{n+1}+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\) |
Input:
Int[(a + b*x + c*x^2)*(1 + (a*x + (b*x^2)/2 + (c*x^3)/3)^n),x]
Output:
a*x + (b*x^2)/2 + (c*x^3)/3 + (a*x + (b*x^2)/2 + (c*x^3)/3)^(1 + n)/(1 + n )
Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[P q, x], r = Expon[Qr, x]}, Simp[Coeff[Qr, x, r]/(q*Coeff[Pq, x, q]) Subst[ Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x, r]*D [Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] & & PolyQ[Qr, x]
Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(x a +\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}+\frac {\left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+x a \right )^{1+n}}{1+n}\) | \(43\) |
default | \(x a +\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}+\frac {\left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+x a \right )^{1+n}}{1+n}\) | \(43\) |
risch | \(\frac {c \,x^{3}}{3}+\frac {b \,x^{2}}{2}+x a +\frac {x \left (2 c \,x^{2}+3 b x +6 a \right ) \left (\frac {1}{3}\right )^{n} \left (\frac {1}{2}\right )^{n} {\left (x \left (2 c \,x^{2}+3 b x +6 a \right )\right )}^{n}}{6 n +6}\) | \(63\) |
norman | \(x a +\frac {a x \,{\mathrm e}^{n \ln \left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+x a \right )}}{1+n}+\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}+\frac {b \,x^{2} {\mathrm e}^{n \ln \left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+x a \right )}}{2+2 n}+\frac {c \,x^{3} {\mathrm e}^{n \ln \left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+x a \right )}}{3+3 n}\) | \(107\) |
parallelrisch | \(\frac {2 x^{3} {\left (\frac {x \left (2 c \,x^{2}+3 b x +6 a \right )}{6}\right )}^{n} c^{2}+2 x^{3} c^{2} n +2 c^{2} x^{3}+3 x^{2} {\left (\frac {x \left (2 c \,x^{2}+3 b x +6 a \right )}{6}\right )}^{n} b c +3 b c n \,x^{2}+3 b c \,x^{2}+6 x {\left (\frac {x \left (2 c \,x^{2}+3 b x +6 a \right )}{6}\right )}^{n} a c +6 x a c n +6 x a c -18 a b n -18 a b}{6 c \left (1+n \right )}\) | \(141\) |
orering | \(\frac {x \left (2 c \,x^{2}+3 b x +6 a \right ) \left (6 c^{2} n \,x^{4}+12 b c n \,x^{3}+10 c^{2} x^{4}+12 a c n \,x^{2}+6 x^{2} b^{2} n +20 b c \,x^{3}+12 a b n x +24 x^{2} a c +9 b^{2} x^{2}+6 a^{2} n +18 a b x +6 a^{2}\right ) \left (1+\left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+x a \right )^{n}\right )}{36 \left (1+n \right ) \left (c \,x^{2}+b x +a \right )^{2}}-\frac {x^{2} \left (2 c \,x^{2}+3 b x +6 a \right )^{2} \left (\left (2 c x +b \right ) \left (1+\left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+x a \right )^{n}\right )+\frac {\left (c \,x^{2}+b x +a \right )^{2} \left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+x a \right )^{n} n}{\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+x a}\right )}{36 \left (1+n \right ) \left (c \,x^{2}+b x +a \right )^{2}}\) | \(259\) |
Input:
int((c*x^2+b*x+a)*(1+(1/3*c*x^3+1/2*b*x^2+x*a)^n),x,method=_RETURNVERBOSE)
Output:
x*a+1/2*b*x^2+1/3*c*x^3+(1/3*c*x^3+1/2*b*x^2+x*a)^(1+n)/(1+n)
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.44 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\frac {2 \, {\left (c n + c\right )} x^{3} + 3 \, {\left (b n + b\right )} x^{2} + {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )} {\left (\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x\right )}^{n} + 6 \, {\left (a n + a\right )} x}{6 \, {\left (n + 1\right )}} \] Input:
integrate((c*x^2+b*x+a)*(1+(a*x+1/2*b*x^2+1/3*c*x^3)^n),x, algorithm="fric as")
Output:
1/6*(2*(c*n + c)*x^3 + 3*(b*n + b)*x^2 + (2*c*x^3 + 3*b*x^2 + 6*a*x)*(1/3* c*x^3 + 1/2*b*x^2 + a*x)^n + 6*(a*n + a)*x)/(n + 1)
Timed out. \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\text {Timed out} \] Input:
integrate((c*x**2+b*x+a)*(1+(a*x+1/2*b*x**2+1/3*c*x**3)**n),x)
Output:
Timed out
Time = 0.15 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.66 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x + \frac {{\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )} e^{\left (n \log \left (2 \, c x^{2} + 3 \, b x + 6 \, a\right ) + n \log \left (x\right )\right )}}{3^{n + 1} 2^{n + 1} n + 3^{n + 1} 2^{n + 1}} \] Input:
integrate((c*x^2+b*x+a)*(1+(a*x+1/2*b*x^2+1/3*c*x^3)^n),x, algorithm="maxi ma")
Output:
1/3*c*x^3 + 1/2*b*x^2 + a*x + (2*c*x^3 + 3*b*x^2 + 6*a*x)*e^(n*log(2*c*x^2 + 3*b*x + 6*a) + n*log(x))/(3^(n + 1)*2^(n + 1)*n + 3^(n + 1)*2^(n + 1))
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.84 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x + \frac {{\left (\frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x\right )}^{n + 1}}{n + 1} \] Input:
integrate((c*x^2+b*x+a)*(1+(a*x+1/2*b*x^2+1/3*c*x^3)^n),x, algorithm="giac ")
Output:
1/3*c*x^3 + 1/2*b*x^2 + a*x + (1/3*c*x^3 + 1/2*b*x^2 + a*x)^(n + 1)/(n + 1 )
Time = 21.84 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.46 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=a\,x+\left (\frac {3\,b\,x^2}{6\,n+6}+\frac {2\,c\,x^3}{6\,n+6}+\frac {6\,a\,x}{6\,n+6}\right )\,{\left (\frac {c\,x^3}{3}+\frac {b\,x^2}{2}+a\,x\right )}^n+\frac {b\,x^2}{2}+\frac {c\,x^3}{3} \] Input:
int(((a*x + (b*x^2)/2 + (c*x^3)/3)^n + 1)*(a + b*x + c*x^2),x)
Output:
a*x + ((3*b*x^2)/(6*n + 6) + (2*c*x^3)/(6*n + 6) + (6*a*x)/(6*n + 6))*(a*x + (b*x^2)/2 + (c*x^3)/3)^n + (b*x^2)/2 + (c*x^3)/3
Time = 0.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.62 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx=\frac {x \left (6 \left (2 c \,x^{3}+3 b \,x^{2}+6 a x \right )^{n} a +3 \left (2 c \,x^{3}+3 b \,x^{2}+6 a x \right )^{n} b x +2 \left (2 c \,x^{3}+3 b \,x^{2}+6 a x \right )^{n} c \,x^{2}+6 \,6^{n} a n +6 \,6^{n} a +3 \,6^{n} b n x +3 \,6^{n} b x +2 \,6^{n} c n \,x^{2}+2 \,6^{n} c \,x^{2}\right )}{6 \,6^{n} \left (n +1\right )} \] Input:
int((c*x^2+b*x+a)*(1+(a*x+1/2*b*x^2+1/3*c*x^3)^n),x)
Output:
(x*(6*(6*a*x + 3*b*x**2 + 2*c*x**3)**n*a + 3*(6*a*x + 3*b*x**2 + 2*c*x**3) **n*b*x + 2*(6*a*x + 3*b*x**2 + 2*c*x**3)**n*c*x**2 + 6*6**n*a*n + 6*6**n* a + 3*6**n*b*n*x + 3*6**n*b*x + 2*6**n*c*n*x**2 + 2*6**n*c*x**2))/(6*6**n* (n + 1))